The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about the EMPIRICAL RULE to answer the following.

a)Approximately what percent of men are taller than 69 inches?

b)Approximately what percent of men are between 64 and 66.5 inches?

Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95​% confidence that the sample mean is plus or minus3 IQ points of the true mean. Assume that the standard deviation is 15 and determine the required sample size.

2)

In a survey of

2,416 adults, 1,876 reported that​ e-mails are easy to​ misinterpret, but only 1,231 reported that telephone conversations are easy to misinterpret. Complete parts​ (a) through​ (c) below.

a. Construct a​ 95% confidence interval estimate for the population proportion of adults who report that​ e-mails are easy to misinterpret.

EXERCISES ON DISCRETE DISTRIBUTIONS

6. An exam consists of 12 questions that present four possible answers each. A person, without knowledge about the subject of the exam, answers the random exam questions.
a.What is the probability that you get the right answer when answering a question?
b. Find the probability that such person does not answer any questions well
C. Calculate the probability of correcting a question.
d. Obtain the probability that you answer all the questions correctly.
e. Obtain the probability of answering more than half of the questions correctly

1. Identify the possible values of each of the 3 variables in this dataset and describe what information each of the 3 variables tells us about the data

upper quarter. 2. During the years 1998–2012, a total of 29 earthquakes
of magnitude greater than 6.5 have occurred in Papua New Guinea. Assume that the time
spent waiting between earthquakes is
exponential. Do in R (Practice and check
with calculator) a. What is the probability that the next earthquake
occurs within the next three months?
b. Given that six months has passed without an
earwhat is the pr thquake in Papua New Guinea, obability that the next three
months will be free of earthquakes? c. What is the probability of zero
earthquakes occurring in 2014? d. What is the probability that at least two
earthquake distributed with a mean of 100 and

Let ? be the sample space of an experiment and let ℱ be a collection of subsets of ?.

a) What properties must ℱ have if we are to construct a probability measure on (?,ℱ)?

b) Assume ℱ has the properties in part (a). Let ? be a function that maps the elements of ℱ onto ℝ such that

i) ?(?) ≥ 0 , ∀ ? ∈ ℱ ii) ?(?) = 1 and iii) If ?1 , ?2 … are disjoint subsets in ℱ then ?(⋃ ??) = ∞ ?=1 ∑ ?(??) ∞ ?=1 . Show that 0 ≤ ?(?) ≤ 1, ∀? ∈ ℱ.

c) Is every subset of ? necessarily an event? Explain briefly. Rigorous definitions are not necessary.

d) Assume ℱ has the properties in part (a). Let ? and ? be any two subsets of ? that are elements of ℱ.

i) Show that (? ∩ ?) ∈ ℱ.

ii) Show that (? ∖ ?) ∈ ℱ, where (? ∖ ?) is the set of outcomes that are in ? but not in ?.

iii) Show that (? △ ?) ∈ ℱ, where (? △ ?) is the set of outcomes that are either in ? or in ? but not in both.

iv) Let ?1 , ?2 , ?3 … be elements of ℱ. Show that ⋂ ?? ∞ ?=1 ∈ ℱ

Joe can choose to take the freeway or not for going to work.

There is a 0.4 chance for him to take the freeway. If he chooses freeway, Joe is late to work with probability 0.3; if he avoids the freeway, he is late

with probability 0.1

Given that Joe was early, what is the probability that he took freeway?

A researcher believes that in recent years women have been getting taller. Ten years ago, the average height of young adult women living in his city was 63 inches. The standard deviation is not known. He randomly samples eight young adult women currently residing in the city and measures their heights.

The sample data obtained is: [64, 66, 68, 60, 62, 65, 66, 63]. Use a significance level of alpha=0.05 to test if women are now taller.

A state highway goes through a small town where the posted speed limit drops down to 40MPH, but which out of town drivers don’t observe very carefully. Based on historical data, it is known that passenger car speeds going through the city are normally distributed with a mean of 47 mph and a standard deviation of 4MPH. Truck speeds are found to be normally distributed with a mean of 45MPH and a standard deviation of 6MPH. The town installed a speed camera and wants to set a threshold for triggering the camera to issue citations. If the camera is triggered, the driver is mailed a flat $50 ticket for cars and a flat $75 for trucks. On average 100 cars and 25 trucks go through the city in a day.

1. If the town sets the camera triggering speed at 50MPH, how much revenue will it make in a month (assume a month has 30 days)
2. The town wants to set the triggering speed at a value such that the fastest 10% of truck drivers get ticketed. At what value should they set the trigger?
3. At this trigger value, what percentage of cars are ticketed and what is the monthly revenue for the city?

In a multiple choice exam, there are 7 questions and 4 choices for each question (a, b, c, d). Nancy has not studied for the exam at all and decides to randomly guess the answers. What is the probability that: (please round all answers to four decimal places)

a) the first question she gets right is question number 3?

b) she gets all of the questions right?

c) she gets at least one question right?

A) Suppose that the mean and standard deviation of the scores on a statistics exam are 89.2 and 6.49, respectively, and are approximately normally distributed. Calculate the proportion of scores below 77.

B)

When students use the bus from their dorms, they have an average commute time of 8.974 minutes with standard deviation 3.1959 minutes. Approximately 66.9% of students reported a commute time less than how many minutes? Assume the distribution is approximately normal.

C)

The revenue of 200 companies is plotted and found to follow a bell curve. The mean is $637.485 million with a standard deviation of $27.6736 million. Would it be unusual for a randomly selected company to have a revenue above $687.08 million?

Consider the following sample data:

a. Calculate the range.

b. Calculate MAD. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

c. Calculate the sample variance. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

d. Calculate the sample standard deviation. (Round your intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

a) In a small country, the probability that a person will die from a certain respiratory infection is 0.004. Let ? be the random variable representing the number of persons infected who will die from the infection. A random sample of 2000 persons with this disease is chosen.

(i) Determine the exact distribution of ? and state TWO reasons why it was chosen? [4 marks]

(ii) State the values of ?(?) and ???(?). [2 marks]

(iii) Using a suitable approximate distribution, find the probability that fewer than 5 persons will die from the infection. (Do not use the exact distribution in part (i)). [4 marks]

The Wilson family had 9 children. Assuming that the probability of a child being a girl is 0.5, find the probability that the Wilson family had: at least 2 girls? at most 3 girls? Round your answers to 4 decimal places.

A bag contains 12 balls of the same shape and size.

Of these, 9 balls are blue, and the remaining 3 balls are red.

Suppose that you do the following iterative random experiment: In each iteration, 5 balls are removed randomly (without replacement) from the bag, in such a way that any 5 balls in the bag are equally likely to be the 5 balls that are removed. After doing this, you check whether among the 5 removed balls there are exactly 2 red balls. If so, then you STOP. Otherwise, you replace the 5 balls back into the bag, shake the bag up (to make sure it is randomly mixed again), and repeat the same experiment: random sample 5 balls from the bag, and check whether you have taken out exactly 2 red balls.

You repeat this until the process STOPs (i.e., when the 5 removed balls in some iteration contain exactly 2 red balls among them).

What is the expected number of times that you will sample 5 balls from this bag, in the above random experiment?